A Nontrivial Ultraviolet Fixed Point and Stability of Three-Dimensional O(N) Model with Fermions and Bosons

Abstract

A three-dimensional O(N) model with both fermions and scalar bosons are studied extensively up to the second order in the 1/ N expansion. Fermionic interactions work towards stabilizing the vacuum as seen from the Gaussian variational method. Vacuum stability requires a constraint among dimensionless coupling constants. The effective potential is calculated up to the second order. Renormalizing the effective potential, we calculate the renormalization-group functions. There are some fixed points. The renormalization-group flow diagram shows that there exists an ultraviolet stable fixed point compatible with the vacuum stability constraint. The fixed point corresponding to that in the supersymmetric limit is seen to be unstable unless there is an exact supersymmetry from the beginning. § 1. IntroductionIt is an important problem in relativistic quantum field theory what are criteria for choosing non-trivial and mathematically consistent theories. It has been claimed recently I) that the simplest (rjJ4)d theory is trivial for d>4 (and possibly also for d=4 2 )) in the limit of infinite ultraviolet cutoff. Are all non-asymptotically-free theories trivial?The l}(rjJ6)3 theory is also renormalizable and non-asymptotically-free. The l/N analyses of its O(N) symmetric version have revealed existence of the nontrivial ultraviolet fixed point (UVFP) at l} = 192, when the coupling is defined by (6N2)-Il) ((P) 3.3)_5)It was observed, however, by a variational method that the theory becomes unstable for l} >(47r)2 and l}<O.6) Although l}=192>(47r)2, it was argued in Ref. 6) that a new UVFP exists at l} = (47r)2 with which a new phase is associated. But it has no stable vacuum, either.7),8)When an N -component fermi field is introduced in addition to the N -component scalar field, there appear other types of renormalizable and 1/ N -calculable interactions in three dimensions. 9 )-II)As is well-known in supersymmetric (SUSY) theories, such fermionic interactions are expected to improve the ultraviolet behavior of the system. It is the aim of this work to study effects of such new interactions on the stability problem and to discuss the ultraviolet behavior of the model up to the next-to-leading 1/ N (NLN) order.We first analyse the system using the Gaussian variational method in the next section. We find that fermionic interactions drastically improve stability of the vacuum. Stability requires a constraint among coupling constants. In § 3, we calculate the effective potential up to the NLN order following Cornwall-lackiw-Tomboulis. 12 ) Renormalization of divergences of the effective potential is performed in § 4. Renormalization-group flow and fixed-point structure are studied in § 5. A stable UVFP is found to exist in a domain compatible with the constraint. The final section is devoted to conclusions.

Finite N origin of the Bardeen-Moshe-Bander phenomenon and its extension at N=∞ by singular fixed points

“…Many aspects and extensions of the BMB phenomenon have been explored, some involving fermions and supersymmetry [3,4,[11][12][13][14]. However, despite intensive efforts, at least two major questions have remained open: Is there a finite N extension of the BMB line of FPs?…”

Phases of supersymmetric O(N) theories

“…The O(N ) symmetric Wess-Zumino model with a microscopic (Φ 2 ) 2 superpotential is determined by only two renormalized parameters and critical and tricritical theories are the same. Its phase structure has attracted some attention in the past [6][7][8][9][10][11][12][13]. In the limit of infinitely many superfields, four different phases have been observed [6,7], including peculiar degenerate O(N ) symmetric phases with several mass scales.…”